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G = D1047C2order 416 = 25·13

The semidirect product of D104 and C2 acting through Inn(D104)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1047C2, C52.35D4, C8.17D26, C4.20D52, Dic527C2, C22.1D52, C52.30C23, D52.7C22, C104.17C22, Dic26.6C22, (C2×C8)⋊4D13, (C2×C104)⋊6C2, C131(C4○D8), C104⋊C27C2, (C2×C26).18D4, C2.13(C2×D52), C26.11(C2×D4), (C2×C4).81D26, D525C21C2, (C2×C52).99C22, C4.28(C22×D13), SmallGroup(416,125)

Series: Derived Chief Lower central Upper central

C1C52 — D1047C2
C1C13C26C52D52D525C2 — D1047C2
C13C26C52 — D1047C2
C1C4C2×C4C2×C8

Generators and relations for D1047C2
 G = < a,b,c | a104=b2=c2=1, bab=a-1, ac=ca, cbc=a52b >

Subgroups: 512 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C13, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], D13 [×2], C26, C26, C4○D8, Dic13 [×2], C52 [×2], D26 [×2], C2×C26, C104 [×2], Dic26 [×2], C4×D13 [×2], D52 [×2], C13⋊D4 [×2], C2×C52, C104⋊C2 [×2], D104, Dic52, C2×C104, D525C2 [×2], D1047C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C4○D8, D26 [×3], D52 [×2], C22×D13, C2×D52, D1047C2

Smallest permutation representation of D1047C2
On 208 points
Generators in S208
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 165)(2 164)(3 163)(4 162)(5 161)(6 160)(7 159)(8 158)(9 157)(10 156)(11 155)(12 154)(13 153)(14 152)(15 151)(16 150)(17 149)(18 148)(19 147)(20 146)(21 145)(22 144)(23 143)(24 142)(25 141)(26 140)(27 139)(28 138)(29 137)(30 136)(31 135)(32 134)(33 133)(34 132)(35 131)(36 130)(37 129)(38 128)(39 127)(40 126)(41 125)(42 124)(43 123)(44 122)(45 121)(46 120)(47 119)(48 118)(49 117)(50 116)(51 115)(52 114)(53 113)(54 112)(55 111)(56 110)(57 109)(58 108)(59 107)(60 106)(61 105)(62 208)(63 207)(64 206)(65 205)(66 204)(67 203)(68 202)(69 201)(70 200)(71 199)(72 198)(73 197)(74 196)(75 195)(76 194)(77 193)(78 192)(79 191)(80 190)(81 189)(82 188)(83 187)(84 186)(85 185)(86 184)(87 183)(88 182)(89 181)(90 180)(91 179)(92 178)(93 177)(94 176)(95 175)(96 174)(97 173)(98 172)(99 171)(100 170)(101 169)(102 168)(103 167)(104 166)
(105 157)(106 158)(107 159)(108 160)(109 161)(110 162)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 169)(118 170)(119 171)(120 172)(121 173)(122 174)(123 175)(124 176)(125 177)(126 178)(127 179)(128 180)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 188)(137 189)(138 190)(139 191)(140 192)(141 193)(142 194)(143 195)(144 196)(145 197)(146 198)(147 199)(148 200)(149 201)(150 202)(151 203)(152 204)(153 205)(154 206)(155 207)(156 208)

G:=sub<Sym(208)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,165)(2,164)(3,163)(4,162)(5,161)(6,160)(7,159)(8,158)(9,157)(10,156)(11,155)(12,154)(13,153)(14,152)(15,151)(16,150)(17,149)(18,148)(19,147)(20,146)(21,145)(22,144)(23,143)(24,142)(25,141)(26,140)(27,139)(28,138)(29,137)(30,136)(31,135)(32,134)(33,133)(34,132)(35,131)(36,130)(37,129)(38,128)(39,127)(40,126)(41,125)(42,124)(43,123)(44,122)(45,121)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106)(61,105)(62,208)(63,207)(64,206)(65,205)(66,204)(67,203)(68,202)(69,201)(70,200)(71,199)(72,198)(73,197)(74,196)(75,195)(76,194)(77,193)(78,192)(79,191)(80,190)(81,189)(82,188)(83,187)(84,186)(85,185)(86,184)(87,183)(88,182)(89,181)(90,180)(91,179)(92,178)(93,177)(94,176)(95,175)(96,174)(97,173)(98,172)(99,171)(100,170)(101,169)(102,168)(103,167)(104,166), (105,157)(106,158)(107,159)(108,160)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)(137,189)(138,190)(139,191)(140,192)(141,193)(142,194)(143,195)(144,196)(145,197)(146,198)(147,199)(148,200)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206)(155,207)(156,208)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,165)(2,164)(3,163)(4,162)(5,161)(6,160)(7,159)(8,158)(9,157)(10,156)(11,155)(12,154)(13,153)(14,152)(15,151)(16,150)(17,149)(18,148)(19,147)(20,146)(21,145)(22,144)(23,143)(24,142)(25,141)(26,140)(27,139)(28,138)(29,137)(30,136)(31,135)(32,134)(33,133)(34,132)(35,131)(36,130)(37,129)(38,128)(39,127)(40,126)(41,125)(42,124)(43,123)(44,122)(45,121)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106)(61,105)(62,208)(63,207)(64,206)(65,205)(66,204)(67,203)(68,202)(69,201)(70,200)(71,199)(72,198)(73,197)(74,196)(75,195)(76,194)(77,193)(78,192)(79,191)(80,190)(81,189)(82,188)(83,187)(84,186)(85,185)(86,184)(87,183)(88,182)(89,181)(90,180)(91,179)(92,178)(93,177)(94,176)(95,175)(96,174)(97,173)(98,172)(99,171)(100,170)(101,169)(102,168)(103,167)(104,166), (105,157)(106,158)(107,159)(108,160)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)(137,189)(138,190)(139,191)(140,192)(141,193)(142,194)(143,195)(144,196)(145,197)(146,198)(147,199)(148,200)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206)(155,207)(156,208) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,165),(2,164),(3,163),(4,162),(5,161),(6,160),(7,159),(8,158),(9,157),(10,156),(11,155),(12,154),(13,153),(14,152),(15,151),(16,150),(17,149),(18,148),(19,147),(20,146),(21,145),(22,144),(23,143),(24,142),(25,141),(26,140),(27,139),(28,138),(29,137),(30,136),(31,135),(32,134),(33,133),(34,132),(35,131),(36,130),(37,129),(38,128),(39,127),(40,126),(41,125),(42,124),(43,123),(44,122),(45,121),(46,120),(47,119),(48,118),(49,117),(50,116),(51,115),(52,114),(53,113),(54,112),(55,111),(56,110),(57,109),(58,108),(59,107),(60,106),(61,105),(62,208),(63,207),(64,206),(65,205),(66,204),(67,203),(68,202),(69,201),(70,200),(71,199),(72,198),(73,197),(74,196),(75,195),(76,194),(77,193),(78,192),(79,191),(80,190),(81,189),(82,188),(83,187),(84,186),(85,185),(86,184),(87,183),(88,182),(89,181),(90,180),(91,179),(92,178),(93,177),(94,176),(95,175),(96,174),(97,173),(98,172),(99,171),(100,170),(101,169),(102,168),(103,167),(104,166)], [(105,157),(106,158),(107,159),(108,160),(109,161),(110,162),(111,163),(112,164),(113,165),(114,166),(115,167),(116,168),(117,169),(118,170),(119,171),(120,172),(121,173),(122,174),(123,175),(124,176),(125,177),(126,178),(127,179),(128,180),(129,181),(130,182),(131,183),(132,184),(133,185),(134,186),(135,187),(136,188),(137,189),(138,190),(139,191),(140,192),(141,193),(142,194),(143,195),(144,196),(145,197),(146,198),(147,199),(148,200),(149,201),(150,202),(151,203),(152,204),(153,205),(154,206),(155,207),(156,208)])

110 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D13A···13F26A···26R52A···52X104A···104AV
order1222244444888813···1326···2652···52104···104
size1125252112525222222···22···22···22···2

110 irreducible representations

dim111111222222222
type+++++++++++++
imageC1C2C2C2C2C2D4D4D13C4○D8D26D26D52D52D1047C2
kernelD1047C2C104⋊C2D104Dic52C2×C104D525C2C52C2×C26C2×C8C13C8C2×C4C4C22C1
# reps1211121164126121248

Matrix representation of D1047C2 in GL2(𝔽313) generated by

610
0195
,
0195
610
,
10
0312
G:=sub<GL(2,GF(313))| [61,0,0,195],[0,61,195,0],[1,0,0,312] >;

D1047C2 in GAP, Magma, Sage, TeX

D_{104}\rtimes_7C_2
% in TeX

G:=Group("D104:7C2");
// GroupNames label

G:=SmallGroup(416,125);
// by ID

G=gap.SmallGroup(416,125);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,50,579,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^104=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^52*b>;
// generators/relations

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